The present invention relates to a piezo-electric resonator comprising at least one quartz crystal in the form of a rectangular thin plate, of which the length l is directed according to an axis X', the width w according to an axis Y' and the thickness t according to an axis Z', and vibrating in a contour mode.
A contour mode resonator is realized in the form of a thin plate with displacement in the plane of the plate. The thickness has to be sufficiently small, so that the forces of inertia caused by the displacement outside the plane of the plate, due to crossed elastic constants, have a negligible influence on the energy of deformation. The shape which is most usually adopted is the rectangle, which implies the existence of four geometrical parameters:
two angles of cut for defining the direction of the normal to the plate, PA1 one angle of cut for defining the orientation of the sides of the rectangle in the plane of the plate, and PA1 a dimensional ratio between the sides of the rectangle. PA1 (a) Temperature coefficient of the first order .alpha. zero, controlled by a cutting angle and not dependent to any critical degree on the dimensional ratio of the crystal plate. PA1 (b) Temperature coefficients of higher orders which are as small as possible, with in particular the possibility of cancelling out the temperature coefficient of the second order .beta.. PA1 (c) Sufficient piezo-electric coupling and fairly high motional capacitance C.sub.1 for assuring an acceptable impedance level. PA1 (d) Piezo-electric coupling of all the other modes sufficiently small for fulfilling the condition: ##EQU1## where Q represents the quality factor. This condition guarantees that only the optimised mode is excited by the oscillator. PA1 (e) Use of a substrate obtained by a single crystallographic rotation, of which the influence on the temperature coefficient of the first order .alpha. should not be great, for example, less than 5.10.sup.-6 /.degree.C. per degree of angle. PA1 (f) A second rotation, made about the normal to the plate, does not have to be critical and tolerances of the order of .+-.5' should be acceptable.
For a resonator to be of practical interest, it is necessary for its temperature coefficient of the first order to be close to 0 (for example, .+-.1.10.sup.-6 /.degree.C.) at the mean temperature of use.
Available on the market are quartz crystals of different cuts, among which the GT-cut is that which provides the most favorable thermal properties. What is involved here is a rectangular plate, of which the dimensional ratio is equal to 0.86, and which is obtained by a rotation about the electrical axis X of the crystal, followed by a rotation of .+-.45.degree. about the normal. The GT-cut resonator oscillates in accordance with a contour mode and more particularly in a width-extensional mode. The frequency temperature coefficients of the first order and of the second order are zero, and the coefficient of the third order is very small. The inconvenience arising from this cut is due to the fact that the thermal properties of the resonator are dependent in a critical manner on the dimensional ratio of the plate.
By way of example, as regards a GT-cut quartz crystal, the temperature coefficient of the first order .alpha. is equal to .+-.0.1.10.sup.-6 /.degree.C., the coefficient of the second order .beta. is equal to .+-.1.10.sup.-6 /.degree.C..sup.2 and the coefficient of the third order .gamma. is smaller than 30.10.sup.-12 /.degree.C..sup.3.
However, it is to be noted that, for a relative variation .DELTA.(w/l)/w/l of the dimensional ratio, equal to 1%, the variation of the temperature coefficient of the first order is equal to 2.5.10.sup.-6 /.degree.C. This implies that the temperature coefficient of a GT-cut quartz crystal necessarily has to be adjusted after fitting of the resonator.
Another known section or cut is the DT-cut, which consists of a generally square plate, obtained by a rotation about the electrical axis X and vibrating in face-shear mode. The DT-cut resonators have the advantage, relatively to those of GT-cut, of showing very little sensitivity to the variations of the dimensional ratio. However, their thermal properties are less satisfactory. By way of example, the temperature coefficient of the first order .alpha. is zero, the temperature coefficient of the second order .beta. is equal to -(15 to 20). 10.sup.-9 /.degree.C..sup.2 and the temperature coefficient of the third order .gamma. is approximately equal to 45.10.sup.-12 /.degree.C..sup.3.
AT-cut quartz crystals are also known, which are in the form of a plate obtained by a rotation about the electrical axis X of the crystal. The data concerning these quartz crystals are to be more particularly found in the publication "Quartz vibrators and their applications", by Pierre Vigoureux, edited by "His Majesty's Stationery Office", London, 1950. Two types of AT-cut quartz crystals, oscillating at a frequency of 4 MHz, are marketed at the present time. These are the AT-cut quartz crystal of Nihon Dempa Cie, of which the thermal properties at 25.degree. C. are as follows:
Temperature coefficient of the first order: EQU .alpha.=.+-.0.1.10.sup.-6 /.degree.C. PA0 Temperature coefficient of the second order: EQU .beta.=-4.+-.1.10.sup.-9 /.degree.C..sup.2 PA0 Temperature coefficient of the third order: EQU .gamma.=95.10.sup.-12 /.degree.C..sup.3 PA0 Temperature coefficient of the first order: EQU .alpha.=.+-.0.1.10.sup.-6 /.degree.C. PA0 Temperature coefficient of the second order: EQU .beta.=-11.+-.1.10.sup.-9 /.degree.C..sup.2 PA0 Temperature coefficient of the third order: EQU .gamma.=90.10.sup.-12 /.degree.C..sup.3.
and the AT-cut quartz crystal of Societe Suisse pour l'Industrie Horlogere (SSIH), of which the thermal properties at 25.degree. C. are as follows:
As well as the thermal properties being inferior to those of the GT-cut quartz crystals, the AT-cut quartz crystals have a frequency four times higher for a comparable size. On the other hand, the temperature coefficient of the first order is more sensitive to the differences in values of the angle of cut, e.g. for a variation .DELTA..phi. of the angle .phi., equal to 1.degree., the corresponding variation .DELTA..alpha. of the temperature coefficient of the first order is equal to 4.7.10.sup.-6 /.degree.C. In addition, the AT-cut quartz crystal of Nihon Dempa Cie, has a complicated shape, presenting a bevelling at each end of the bar and an inclination of the lateral faces. This involves the necessity of an individual metallization after complete machining. The AT quartz of SSIH has a considerable length, i.e. of about 11 mm.
An object of the present invention is to overcome the aforementioned disadvantages of the quartz crystals of known cuts and to provide a resonator of the type as initially indicated, which satisfies all the following conditions: